Stability of Shifted Complexes via the Second-Moment Defect of the Up-Laplacian

What the math gives to ML

The paper identifies an exact second-moment defect: the gap between the squared norm of the conjugate degree partition and the squared norm of the nonzero up-Laplacian spectrum equals twice the number of failed elementary shifts. Its stability theorem converts this defect into an explicit upper bound on the number of facet exchanges needed to reach a shifted family, making it a computable structural-distance certificate rather than only an equality characterization. A plausible neural transfer is to regularize sparse attention or routing graphs toward shifted, nested-neighborhood structures that may simplify indexing and improve load regularity. The most practical first experiment is the graph case k=1, where the defect can be computed directly from degrees and edge count without eigendecomposition.